Any one of eight potential successors as party leader would significantly improve the party’s fortunes, according to the ComRes survey for The Independent.
If Foreign Secretary David Miliband or Justice Secretary Jack Straw took over, Labour would even be returned as the largest party in a hung Parliament, the poll found.
With Mr Brown as leader, by contrast, the Tories are on course for a 48-seat majority, based on current voting intentions.
The poll showed Labour support down to 23% - neck-and-neck for the first time with third party the Liberal Democrats - and the Conservatives on 38%.
Labour’s support would increase under any one of eight senior party figures seen as potential candidates for the leadership post-Brown, ComRes found.
Under Jon Cruddas, the left-leaning backbencher, Tory leader David Cameron would still be in line for a Commons majority.
But he would be left short of a majority if either Lord Mandelson, Harriet Harman, Alan Johnson, Ed Balls or Ed Miliband took over as leader.
And with Mr Straw or David Miliband, Labour would be the biggest party. With Mr Straw as leader, voting intentions were 31% Labour, 28% Tory and 21% Lib Dem.
With David Miliband, they were 31% Labour, 28% Tory and 21% Lib Dem.
ComRes telephoned 1,003 British adults between September 25 and 27, weighting data by past vote recall.
Business Secretary Lord Mandelson said the polls reflected the fact that the Prime Minister has taken “quite a beating” due to the recession and expenses scandal, but insisted he could regain his popularity as voters see the results of his efforts to return the economy to health.
He said it had never occurred to him to ask the PM to step aside in favour of another leader who might perform better in the polls and described Mr Brown as “the best man for the job”.
Voters were “far from certain, far from convinced” about David Cameron’s Conservatives and defeat for Labour was not “inevitable”, he insisted.
Lord Mandelson told GMTV: “I think the public have a far from settled view and embrace of the Conservatives as the alternative to Labour. I think people are obviously worried about the economy, about their jobs and the future chances for their children, but I think they are far from certain, far from convinced that the Conservatives present a good alternative.
“The country has had quite a beating over the last year. The Government therefore has had quite a beating with what’s happened with the banks, the recession and all the furore at Westminster about expenses and the Prime Minister is out there on the front line.”
Lord Mandelson said it was too early for voters to judge the outcome of the Government’s approach to the recession as “we are still being tested by events”.
“I think people are saying `We are still coming through the recession, we are not locked into recovery yet, we are going to judge the Government when the time comes by the results of their policies and how well they have guided us through this and what the alternative would have been for the country if the Conservatives were in place’,” he said.
“That judgment will come at the election, it doesn’t come now. I think Gordon Brown will come out of that much more favourably.”
Asked if he ever considered advising the Prime Minister to step down to give Labour a better chance of victory under another leader, Lord Mandelson told BBC1’s Breakfast: “No, it hasn’t occurred to me because I believe he is the best man for the job, because he has very clear and firm ideas about how to lead the country through this very difficult period, he has a very firm grasp of the big picture facing us in this country and also because I know him.
“He is a man who sticks to his guns and I would much rather have a man who sticks to his guns and knows where he is going, rather than some sort of shallow flibbertigibbet like David Cameron.”
Mr Cameron had failed to change his own party and had adopted economic policies which would “jeopardise if not wreck the economy”, said Lord Mandelson.
The Business Secretary said Chancellor Alistair Darling was right to warn Labour that it sometimes seemed like a football team which had lost “the will to live”.
“He needs to give us a good kick up the pants, because what we need to do is fight back,” said Lord Mandelson.
“We are the political underdogs. The polls are clearly saying that, but the election is not now, or tomorrow or the day after.
“Of course we have got to be proud of what we have achieved over the last 12 years and the fightback against the recession over the last year, but when it comes to the election, people will be voting about the future, not the past. We have got to be clear about how we are going to fix those things that are still wrong in Britain.”
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